GATE 2006

Q. 1 - Q. 20 carry one mark each

Column IP. Gauss-Seidel methodQ. Forward Newton-Gauss method

R. Runge-Kutta method

S. Trapezoidal Rule

(II) 1'-1, Q-4, R-3, 5-2(e) 1'-1, Q-3, R-2, 5-4

2(a) (1 + x) e+x

2

(e) (l_x)e+x

Column II1. Interpolation2. Non-linear differential equations

3. Numerical integration

4. Linear algebraic equations

(b) pol, Q-4, R-2, 5·3(d) P-4, Q-l, R-2, 5-3

dy _x2- + 2xy = e with y(O) = 1 isdx

2

(b) (1 + x)e-x

2(d) (1 - x) eX

3. Let x denote a real number. Find out the INCORRECT statement.(a) S = {x: x > 31 represents the set of all real numbers greater than 3(b) S = {x: x2 < OJ represents the empty set(e) S = {x: x EA and x EBj represents the union of set A and set B(d) 5 = {x: a < x < bj represents the set of all real numbers between a and b, where a and bare

real numbers

4. A box contains 20 defective items and 80 non-defective items. If two items are selected atrandom without replacement, what will be the probability that both items are defective?

1(a) -

.5

20(e) 99

1(b) 25

19(d) 495

5. For a circular shaft of diameter d subjected to torque T, the maximum value of the shear stressis

(64T

a)-Ttd3

16T(e) Ttd3

32T(b) Ttd3

8T(d) -3

Ttd

6. For a four-bar linkage in toggle position, the value of mechanical advantage is(n) 0.0 (b) 0.5

(c) 1.0 (d) 00

(a) mX+cX+k(x-y)=O ~ \ J--+ .,'(b) m(x - jj) + e(x - y) +kx = 0 I " L

(e) mi+e(x-y)+kx=O l-~[}-Li--~f--A/\Nwv\r---i(d) m(x- y)+ c(x- y) + k(x - y) = 0 C 777777~1 fA pin-ended column of length L, modulus of elasticity E and second moment of thecross-sectional area I is loaded centrically by a compressive load P. The critical buckling load(Per) is given by

EI(a) Per =-2-

1t errEI

(e) Per = L2

The number of inversions for a slider crank mechanism is(a) 6 (b) 5(e) 4 (d) 3

For a Newtonian fluid(a) Shear stress is proportional to shear strain(b) Rate of shear stress is proportional to shear strain(c) Shear stress is proportional to rate of shear strain(d) Rate of shear stress is proportional to rate of shear strain

11. In a two-dimensional velocity field with velocities u and v along the x and y directions respectively, theconvective acceleration along the x-direction is given by

au au au av(a) u-+v- (b) U-+V-

ax CtJ ax CtJ

av au au au(e)u-+v- (d) v-+u-

ax OtJ ax Oy

12. Dew point temperature is the temperature at which condensation begins when the air is cooledat constant(a) volume (b) entropy(e) pressure (d) enthalpy

13. In a composite slab, the temperature at the interface (Tinter) between two materials is equal tothe average of the temperatures at the two ends. Assuming steady one-dimensional heatconduction, which of the following statements is true about the respective thermalconductivities?

(a) 2k1 = k2

(b) k1 = k2

(e) 2k] = 3k2

(d) k1 = 2k2

14. In a Pelton wheel, the bucket peripheral speed is 10 mis, the water jet velocity is25 m/s and volumetric flow rate of the jet is 0.1 m3 Is. If the jet deflection angle is 1200 and theflow is ideal, the power developed is(a) 7.5 kW

(c) 22.5 kW(b) 15.0 kW(d) 37.5 kW

15. An expendable pattern is used in(a) slush casting(c) centrifugal casting

(b) squeeze casting(d) investment casting

(b) machinability of low carbon steels(c) hardenability of high carbon steels(d) machinability of high carbon steels

17. NC contouring is an example of(a) continuous path positioning(c) absolute positioning

,(b) point-to-point positioning(d) incremental positioning

18. A ring gage is used to measure(a) outside diameter but not roundness(b) roundness but not outside diameter(c) both outside diameter and roundness(d) only external threads

19. Thenumber of customers arriving at a railway reservation counter isPoisson distributed with an arrivalrate of eight customers per hour. The reservation clerk at this counter takes sixminutes per customer onan average with an exponentially distributed service time. The average number of the customers in thequeue will be(a) 3

(c) 4

(b) 3.2

(d) 4.2

20. In an MRP system, component demand is(n) forecasted(b) established by the master production schedule(c) calculated by the MRP system from the master production schedule(d) ignored

21. Eigenvalues of a matrix 5 == [~~Jare 5 and 1.What are the eigenvalues of the matrix 52 = SS?

(a) 1 and 25(c) 5 and 1

(b) 6 and 4(d) 2 and 10

22. Equation of the line normal to function I(x) = (x - 8)2/3 + 1 at P (0,5) is(a) y = 3x - 5 (b) Y = 3x + 5(c) 3y = x + 15 (d) 3y = x-IS

pl323. Assuming i= ~ and t is a real number, f eitdt is

o

(a)../3 .1 (b) ../3 - i!-+1-2 2 2 2

1 . ../3 1 {I ../3](c) -+1- (d) 2"+1 -2

2 2

2x2 -7x+324. IfI( x) = 2 ' then Urn I (x) will be

5x -12x - 9 x--+3

1(a) 3

5(b) 18

2(d) 5

25. Match the items in columns I and II.Column I

P. Singular matrixQ. Non-square matrixR. Real symmetric matrix5. Orthogonal matrix

(a) P-3, Q-l, R-4, 5-2

(c) P-3, Q-2, R-5, 5-4

Column II1. Determinant 1S not defined2. Determinant is always one3. Determinant is zero4. Eigen values are always real5. Eigenvalues are not defined

(b) P-2, Q-3, R-4, 5-1

(d) P-3, Q-4, R-2, 5-12

26. For d ~ + 4 dy + 3y = 3e2x , the particular integral isdx dx

1 2x(a) -e15

(c) 3e2x

GATE: 2008 - 8

27. Multiplication of matrices E and F is G. Matrices E and G are

Eo [~~~ ~~~: ~] and G • [HnWhati, ilie matrix Fl

(b) [:~~e~~~8]001

28. Consider a continuous random variable with probability density function

f(t) = 1 + t for -1 ~ t ~ 0

= 1 - t for 0 ~ t ~ 1The standard deviation of the random variable is

1(a) J3

1(c) 3

29. Match the items in columns I and II.

Column IP.AddendumQ. Instantaneous center of velocityR. Section modulus

5. Prime circle(a) P-4, Q-2, R-..J,S-l(c) P-3, Q-2, R-1, 5-4

1(b) J6

1(d) 6

Column II1. Cam2. Beam3. Linkage

4. Gear(b) P-4, Q-3, R-2, 5-1(d) P-3, Q-4, R-1, 5-2

30. A disk clutch is required to transmit 5 kW at 2000 rpm. The disk has a friction lining withcoefficient of friction equal to 0.25. Bore radius of friction lining is equal to 25 mm. Assumeuniform contact pressure of 1 MPa. The value of outside radius of the friction lining is(a) 39.4 mm (b) 49.5 mm(c) 97.9 mm (d) 142.9 mm

31. Twenty degree full depth involute profiled 19-tooth pinion and 37-tooth gear are in mesh. Ifthe module is 5 mm, the center distance between the gear pair will be(a) 140 mm (b) 150 mm(c) 280 mm (d) 300 mm

32. A cylindrical shaft is subjected to an alternating stress of 100 MPa. Fatigue strength to sustain1000 cycles is 490 MPa. If the corrected endurance strength is 70 MPa, estimated shaft life willbe(a) 1071 cycles(c) 281914 cycles

(b) 15000 cycles(d) 928643 cycles

33. According to Von-Mises' distortion energy theory, the distortion energy under three dimensionalstress state is represented by

1[2 2 2 ](a) 2E 0'1 +0'2 +0'3 -2V(0'10'2 +0'30'2 +0'10'3

1- 2V[ 2 2 2 ](b) 6E 0'1 +0'2 +0'3 +2(0'10'2 +0'30'2 +·0'10'3)

I+V[ 2 2 2 ](c) 3E 0'1 +0'2 +0'3 -(0'10'2 +0'30'2 +0'10'3)

1 [2 2 2 ](d) 3E 0'1 +0'2 +0'3 - V(0'10'2 +0'30'2 +0'10'3)

34. A steel bar of 40 mm x 40 mm square cross-section is subjected to an axial compressive load of200 kN. If the length of the bar is 2 m and E = 200 GPa, the elongation of the bar will be(a) 1.25 mm (b) 2.70 mm(c) 4.05 mm (d) 5.40 mm

35. If Cfis the coefficient of speed fluctuation of a flywheel then the ratio of wmax /wmin willbe

1-2Cf 2-Cf(a) 1+ 2Cf (b) 1+ 2Cf

1+2Cf 2+Cf(c) 1- 2C f (d) 2 - Cf

36. A bar having a cross-sectional area of 700mm 2 is subjected to axial loads at the positionsindicated. The value of stress in the se,gment QR is

P Q R S(a) 40 MPa (b) 50 MPa (c) 70 MPa (d) 120 MPa

37. If a system is in equilibrium and the position of the system depends upon many independentvariables, the principle of virtual work states that the partial derivatives of its total potentialenergy with respect to each of the independent variable must be(a) -1.0 (b) 0(c) 1.0 (d) 00

18. If point A is in equilibrium under the actionof the applied forces, the values of tensionTABand TAC are respectively(a) 520 N and 300 N

(b) 300 Nand 520 N

(c) 450 Nand 150 N(d) 150 N and 450 N

39. Match the items in columns I and II

Column I

P.Higher kinematic pair

Q. Lower kinematic pair

R. Quick return mechanism

S. Mobility of a linkage

1. Grubler's equation

2. Line contact

3. Euler's equation

4. Planer

5.Shaper

6. Surface contact

(a) P-2, Q-6, R-4, 5-3

(c) P-6, Q-2, R-5, 5-3

(b) P-6, Q-2, R-4, 5-1

(d) P-2, Q-6, R-5, 5-1

40. A machine of 250 kg mass is supported on springs of total stiffness 100 kN 1m. Machine has anunbalanced rotating force of 350 N at speed of 3600 rpm. Assuming a damping factor of 0.15,the value of transmissibility ratio is(a) 0.0531(c) 0.0162

(b) 0.9922(d) 0.0028

41. In a four-bar linkage,S denotes the shortest link length, L is the longest link length, P and Q arethe lengths of other two links. At least one of the three moving links will rotate by 3600 if(a) 5 + L :5 P + Q (b) 5 + L > P + Q(c) 5 + P:5 L + Q (d) 5 + P > L +Q

42. A 60 mm long and 6 mm thick fillet weld carries a steady load of 15 kN along the weld. Theshear strength of the weld material is equal to 200 MPa. The factor of safety is(a) 2.4 (b) 3.4

(c) 4.8 (d) 6.8

43. A two-dimensional flow field has velocities along the x and y directions given byu = x2t and v = -2xyt respectively, where t is time. The equation of streamlines is(a) x2 y = constant (b) x y2 = constant(c) x y = constant (d) not possible to determine

44. The velocity profile in fully developed laminar flow in a pipe of diameter D is given byu = uo(~ - 4r2/D2), where r is the radial distance from the center. If the viscosity of the fluid is Il,the pressure drop across a length L of the pipe is

45. A siphon draws water from areservoil:and discharges it out atatmospheric pressure. Assumingideal fluid and the reservoir islarge, the velocity at point P in thesiphon tube is

(a) ~2ghl

(b) ~2ghz

(c) J2g(hz - hI)

(d) ~2g(hz + hI)

..-h I

- - - - - - - - ------- - - - 11-------- - -------=- =-=-:- =-=-:~ - ~=- =-=-:-:- :-:-:- 2--------- --- ------------------- ---- - - - -- --- - - - - - - - - --- -- - - --- - - - - - - - - - - - - --------------------------------------

t -

46. A large hydraulic turbine is to generate 300 kW at 1000 rpm under a head of 40 m. For initialtesting, a 1: 4 scale model of the turbine operates under a head of 10 m. The power generatedby the model (in kW) will be

(a) 2.34

(c) 9.38

(b) 4.68

(d) 18.75

47. The statements concern psychrometric chart.1. Constant relative humidity lines are uphill straight lines to the right2. Constant wet bulb temperature lines are downhill straight lines to the right3. Constant specificvolume lines are downhill straight lines to the right4. Constant enthalpy lines are coincident with constant wet bulb temperature linesWhich of the statements are correct?(a) 2 and 3(c) 1 and 3

(b) 1 and 2(d) 2 and 4

48. A 100 W electric bulb was switched on in a 2.5 m x 3 m x 3 m size thermally insulated roomhaving a temperature of 20°C. The room temperature at the end of 24 hours will be(a) 321°C (b) 341°C(c) 450°C (d) 470°C

49. A thin layer of water in a field is formed after a farmer has watered it. The ambient air conditionsare: temperature 20°C and relative humidity 5%.An extract of steam tables is given below.

Temperature

(0C) -15 -10 -5 0.01 5 10 15 20

Saturation

Pressure (kPa) 0.10 0.26 0.40 0.61 0.87 1.23 1.71 2.34

Neglecting the heat transfer between the water and the ground, the water temperature in thefield after phase equilibrium is reached equals

(a) 1O.3°C(c) -14.5 °C

(b) -1O.3°C(d) 14.5 ° C

50. A horizontal-shaft centrifugal pump liftswater at 65°C. The suction nozzle is onemeter below pump centerline. Thepressure at this point equals 200 kPagauge and velocity is 3 m/s. Steam tablesshow saturation pressure at 65°C is 25 kPa,and specific volume of the saturated liquidis 0.001020 m3/kg. The pump Net PositiveSuction Head (NPSH) in meters is(7) 24(c) 28

I

(b) 26(d) 30

Temperature PSat Specific volume (mj Ikg) Enthalpy (kJ/kg)

(0C) (bar) Saturated Saturated Saturated Saturated

liquid vapour liquid vapour

45 0.09593 0.001010 15.26 188.45 2394.8

342.24 150 0.001658 0.010337 1610.5 2610.5

Specific enthalpy of water in kJ/I<g at 150 bar and 45°C is

(a) 203.60 (b) 200.53(c) 196.38 (d) 188.45

52. Determine the correctness or otherwise of the following Assertion [al and the Reason [rl.

Assertion [a): In a power plant working on a Rankine cycle, the regenerative feed water heatingimproves the efficiency of the steam turbine.

Reason [r): The regenerative feed water heating raises the average temperature of heat additionin the Rankine cycle.(a) Both [a) and [r) are true and [r) is the correct reason for [a](b) Both [a] and [r] are true but [r] is NOT the correct reason for [a](c) Both [a) and [r) are false(d) [a) is false but [r] is true

53. Determine the correctness or otherwise of the following Assertion [a] and the Reason [r].

Assertion [a): Condenser is an essential equipment in a steam power plant.

Reason [r]: for the same mass flow rate and the same pressure rise, a water pump requiressubstantially less power than a steam compressor.(a) Both [a] and [r] are true and [r] is the correct reason for [a](b) Both [a] and [r] are true but [r] is NOT the correct reason for [a](c) [a] is true but [r] is flase(d) [a] is false but [r] is true

E HeatFWork

I ExactJ Inexact

K PathL Point

MTransientNBoundary

(a) F-G-J-K-M (b) E-G-I-K-ME-G-I-K-N F-H-I-K-N

(c) F - H - J - L - N (d) E - G - J - K - NE-H-I-L-M F-H-J-K-M

55. Group I shows different heat addition processes in power cycles. Likewise, Group IIshows different heat removal processes. Group ill lists power cycles. Match items from Groups I,IIandID.Group I . Group II Group III

P. Pressure constant S. Pressure constant 1. Rankine cycleQ. Volume constant T. Volume constant 2. Otto cycleR. Temperature constant U. Temperature constant 3. Carnot Cycle

4. Diesel cycle5. Brayton cycle

(a) P - S - 5R-U-3P-S-lQ~T-2

(c) R-T-3

P-S-1P-T-4Q-S-5

(b) P- S-l

R- U-3P-S-4P-T-2

(d) P - T - 4R-S-3P-S-1P-S-5

56. With an increase in the thickness of insulation around a circular pipe, heat loss to surroundings

due to(a) convection increases, while that due to conduction decreases(b) convection decreases, while that due to conduction increases(c) convection and conduction decreases(d) convection and conduction increases

57. The ultimate tensile strength of a material is 400 MPa and the elongation up to maximum loadis 35%. If the material obeys power law of hardening, then the true stress-true strain relation(stress in MPa) in the plastic deformation range is

(a) a = 540 1;030 (b) a = 775 1;0.30

(c) a = 540 1;0.35 (d) a = 775 1;0.35

58. In a sand casting operation, the total liquid head is maintained constant such that it is equal tothe mould height. The time taken to fill the mould with a top gate is tA' If the same mould isfilled with a bottom gate, then the time taken is tB' Ignore the time required to fill the runnerand frictional effects. Assume atmospheric pressure at the top molten metal surfaces. The relationbetween tA and tB is

59. A 4 mm thick sheet is rolled with 300 mm diameter rolls to reduce thickness without anycharge in its width. The friction coefficient at the work-roll interface is 0.1. The minimum possiblethickness of the sheet that can be produced in a single pass is(a) 1.0 mm (b) 1.5 mm(c) 2.5 mm (d) 3.7 mm

60. In a wire drawing operation, diameter of a steel wire is reduced from 10 mm to8 mm. The mean flow stress of the material is 400 MPa. The ideal force required for drawing(ignoring friction and redundant work) is(a) 4.48 kN (b) 8.97 kN(c) 20.11 kN (d) 31.41 kN

61. Match the items in columns I and II.Column I

P. Wrinkling

Q. Orange peelR. Stretcher strainsS. Earing

(a) P - 6, Q - 3, R - I, S - 2(c) P-2,Q-5,R-3,S-4

1. Yield point elongation

2. Anisotropy3. Large grain size4. Insufficient blank holding force5. Fine grain size6. Excessive blank holding force

(b) P - 4, Q - 5, R - 6, S - 1

(d) P - 4, Q - 3, R - I, 5 - 2

62. In an arc welding process, the voltage and current are 25 V and 300 A respectively. The arc heattransfer efficiency is 0.85 and weldign speed is 8 mm/see. The net heat input (in J/mm) is(a) 64 (b) 797(c) 1103 (d) 79700

63. If each abrasive grain is viewed as a cutting tool, then which of the following represents thecutting parameters in common grinding operations?(a) Large negative rake angle, low shear angle and high cutting speed(b) Large positive rake angle, low shear angle and high cutting speed(c) IJarge negative rake angle, high shear angle and low cutting speed(d) Zero rake angle, high shear angle and high cutting speed

64. Arrange the processes in the increasing order of their maximum material removal rate.

Electrochemical Machining (ECM)

Ultrasonic Machining (USM)

Electron Beam Machining (EBM)

Laser Beam Machining (LBM) and

Electric Discharge Machining (EDM)

(a) USM, LBM, EBM, EDM, ECM (b) EBM, LBM, USM, ECM, EDM

(c) LBM, EBM, USM, ECM, EDM (d) LBM, EBM, USM, EDM, ECM65. Match the items in colunms I and II.

Colunm IP. Charpy testQ. Knoop testR. Spiral testS. Cupping test

(a) P - 4, Q - 5, R - 3, S - 2(c) P - 2, Q - 4, R - 3, S - 5

Colunmll1. Fluidity2. Microhardness3. Formability4. Toughness5. Permeability

(b) P - 3, Q - 5, R - I, S - 4(d) P-4,Q-2,R-l,S-3

66. A manufacturing shop processes sheet metal jobs, wherein each job must pass through twomachines (Ml and M2, in that order). The processing time (in hours) for these jobs is

Machine Jobs

P Q R S T U

M1 15 32 8 27 11 16

M2 6 19 13 20 14 7

The optimal make-span (in hours) of the shop is(a) 120 (b) 115(c) 109 (d) 79

67. Consider the following data for an item.Annual demand: 2500 units per yearOrdering cost:Rs. 100 per order Inventory holding rate: 25% of unitprice

(a) 447

(c) 500(b) 471

(d) ~ 600

68. A firm is required to procure three items (P,Q, andR). The prices quoted for these items (in Rs.) bysuppliers 51, 52 and 53 are given in table. Themanagement policy requires that each item has tobe supplied by only one supplier and one suppliersupply only one item. The minimum total cost (inRs.) of procurement to the firm is

(a) 350(c) 385

(b) 360(d) 395

69. A stockist wishes to optimize the number of perishable items he needs to stock in any month inhis store. The demand distribution for this perishable item is

Demand (in units) 2 3 4 5

Probability 0.10 0.35 0.35 0.20

The stockist pays Rs. 70 for each item and he sells each at Rs. 90. If the stock is left unsold in anymonth, he can sell the item at Rs. 50 each. There is no penalty for unfulfilled demand. Tomaximize the expected profit, the optimal stock level is

(a) 5 units(c) 3 units

(b) 4 units(d) 2 units

Work station I II III IV V VI

Total task time at the workstation 7 9 7 10 9 6(in minutes)

(a) 70% (b) 75%(c) 80% (d) 85%Common Data for Questions 71. 72. 73:In an orthogonal machining operation:Uncut thickness = 0.5 mm Cutting speed = 20 m/min Rake angle = 15 °Width of cut = 5 mm . Chip thickness = 0.7 mmThrust force = 200 N Cutting force = 1200 N

Assume Merchant's theory.

71. The values of shear angle and shear strain, respectively, are(a) 30.3° and 1.98 (b) 30.3° and 4.23(c) 40.2° and 2.97 (d) 40.2° and 1.65

72. The coefficient of friction at the tool-chip interface is(a) 0.23 (b) 0.46

(c) 0.85 (d) 0.95

73. The percentage of total energy dissipated due to friction at the tool-chip interface is(a) 30% (b) 42%

(c) 58% (d) 70%

Aplanetary gear train has four gears and one carrier. Angular velocities of the gears are 00 1,00 2,00 3 and004' respectively. The carrier rotates with angular velocity 005,

Gear 24.H

Gcar 320T

Gcar440T

74. What is the relation between the angular velocities of Gear 1 and Gear 4 ?

(;car II.~T

(01 - (05(a)

(04 -(05=6

(04-(05(b)

<1'1- (05=6

t '.~\~...

(01 -(02

=- (~)(c)(04 -(03

(02 -(05 8(d)

(04 -(05 - 9

75. For (01 = 60 rpm clockwise (cw) when looked from thp ~eft,what is the angular velocity of thecarrier and its direction so that Gear 4 rotates in cJunterclockwise (ccw) direction at twice theangular velocity of Gear 1 when looked from the left?(a) 130 rpm, cw (b) 223 rpm, ccw(c) 256 rpm, cw (d) 156 rpm, ccw

Statement for Linked Answer Questions 76 & 77:A simply supported beam of span length 6m and 75mm diameter carries a uniformly distributedload of 1.5 kN / m

76. What is the maximum value of bending moment?(a) 9 kNm (b) 13.5 kNm

(c) 81 kNm (d) 125 kNm

77. What is the maximum value of bending stress?(a) 162.98 MPa (b) 325.95 MPa(c) 625.95 MPa (d) 651.90 MPa

A vibratory system consists of a mass 12.5 kg, a spring of stiffness 1000 N/m, and a dashpotwith damping coefficient of 15 Ns/m.

78. The value of critical damping of the system is(a) 0.223 Ns/m (b) 17.88 Ns/m(c) 71.4 Ns/m (d) 223.6 Ns/m

79. The value of logarithmic decrement is

(a) 1.35

(c) 0.68

(b) 1.32

(d) 0.66

A football was inflated to a gauge pressure of 1 bar when the ambient temperature was15°C. When the game started next day, the air temperature at the stadium was 5 0C.Assume that the volume of the football remains constant at 2500 cm3•

80. The amount of heat lost by the air in the football and the gauge pressure of air in the football atthe stadium respectively equal(a) 30.6 J, 1.94 bar (b) 21.8 J, 0.93 bar(c) 61.1 J, 1.94 bar (d) 43.7 J, 0.93 bar

81. Gauge pressure of air to which the ball must have been originally inflated so that it wouldequal 1 bar gauge at the stadium is(a) 2.23 bar(c) 1.07 bar

(b) 1.94 bar(d) 1.00 bar

A smooth flat plate with a-sharp leading edge is placed ~along a gas stream flowing _

atU= lOm/s.

Statement for Linked Answer Questions 82 & 83:qr--------~---------------I,,IIII,I

flat plateThe thickness of the boundary layer at section r-s is 10 mm, the breadth of the plate is 1m (intothe paper) and the density of the gas p = 1.0 kg/m3. Assume that the boundary layer is thin,

two-dimensional, and follows a linear velocity distribution, u = U(y /0), at the section r-s, wherey is the height from plate.

82. The mass flow rate (in kg/s) across the section q-r is(a) zero (b) 0.05

(c) 0.10 (d) 0.15

83. The integrated drag force (in N) on the plate, between p-s, is(a) 0.67 (b) 0.33

(c) 0.17 (d) zero

Task Predecessor Expected task time Variance of the task time

(in days) (in days2)

a - 30 25b a 40 64c a 60 81

d b 25 9

e b,c 45 36f d,e 20 9

84. The expected completion time of the project is(II) 238 days (b) 224 days(c) 171 days (d) 155 days

85. The standard deviation of the critical path of the project is

(II) Jl5l days

(c) J200 days

(b) J155 days

(d) J238 days

..GATE: 2006 -18 MECHANICAL ENGINEERING

ANSWERS

1. (d) 2. (b) 3. (c) 4. (d) 5. (b) 6. (d) 7. (c) 8. (d) 9. (c) 10. (c)

11. (a) 12. (c) 13. (d) 14. (c) 15. (d) 16. (d) 17. (a) 18. (c) 19. (b) 20. (c)

21. (a) 22. (b) 23. (a) 24. (b) 25. (a) 26. (b) 27. (c) 28. (*) 29. (b) 30. (a)

31. (a) 32. (~J 33. (c) 34. (a) 35. (d) 36. (a) 37. (b) 38. (a) 39. (d) 40. (d)

41. (a) 42. (b) 43. (a) 44. (d) 45. (c) 46. (QL 47. (a) 48. (c) 49. (*) 50. (a)

51.() 52. (a) 53. (b) 54. (d) 55. (a) 56. (a) 57. (c) 58. (c) 59. (c) 60.()61. (d) 62. (b) 63. (a) 64. (c) 65. (d) 66. (b) 67. (a) . 68. (c) 69. (c) 70. (c)

71. (d) 72. (b) 73. (a) 74. (a) 75. (d) 76. (a) 77. (a) 78. (d) 79. (d) 80. (d)

81. (c) 82. (b) 83. (c) 84. (d) 85. (a)

I EXPLANATIONS Idy 2

First order equation, dx + Py = Q, where P = 2x and Q = e-x

+JPdx +X2e = e

2 dy 2eX _ + (2xeX )y

dx

d 2

dx (yeX) = 1

2yeX = x+C

=> C = 1

2.. Y = (1 +x)e-X

3. IX: x E A and x E BI represents the union of set A and set B.The abov'2 symbol denotes intersection of set 1\and set B. Therefore the statement is incorrect

N(Both defective)P (Both defective) = S I

amp e space19

495

T = .!!-d3 x T or16

16TT= --

1td3

Load (output force)Effort (input force).

For a four bar linkage in toggle position, effort = 0

. . Mechanical Advantage = 00

7. Y -.:I

d2x (dX dY)

m dt2 + C dt - dt + kx = 0

This is the differential equation governing the above vibrating system.

8. The critical buckling load (Per) for a column having length L, modulus of elasticity E andsecond moment of cross sectional area I is loaded centrically.

Condition: Both ends are having pin joint. i.e., hingedso n = 1

10. For a Newtonian fluid, shear stress is proportional to rate of shear strain.

11. Two dimensional velocity field with velocities u, v and along x and Y direction.

au au au au=u-+v-+w-+-ox Oy 02 at

Convective temporalacceleration acceleration

auSince, - = a for 2-dimensional field, therefore02

C . I' au auonvechve acce erahon = u-+v-ox Oy

Constant pressureline

~fTdpt ..

sAir is cooled at constant pressure to make unsaturated air to saturated one.

_ -kl A (T.. _ T ) -k2A(T2 - Tinter)Q - 2b mter I = b

T1 +T2Tinter = 2

., k1 = 2k214. From velocity triangle,

Power developed = f Q(VwI +Vwz) x U = 22.5 KW

Expendable pattern is used in Investment casting.Machinability of high carbon steels.NC contouring is an example of continuos path positioning.Ring gage is used to measure both outside diameter and roundness.

').2 8x8Average number of customer in queue = /1(/1- A) = 10 x (10 _ 8) = 3.2

In a MRP system, component demand is calculated by the MRP system from the MasterProduction Schedule.

then for 52 matrix, the eigen values will be ').~, ').~, ').~, .

For 5 matrix, if eigen values are 1 and 5then for 52 matrix, the eigen values are 1 and 25

22. Given, f(x) = (x - 8)2/3 + 1

mnonnal x mtangent = - 1

dy -1where, mtangent = dx and mnonnal = (dy / dx)

dy = df(x) = ~ (x _ 8r1/3dx dx 3

-3 .mnormal = 2 (x - 8)1/3 or [mnonnal](O,S) = 3

:. Equation of normal is, (y - Yl) = mnonnal(x - Xl)

1. 2x2 -7x+3

1= Im-----x~3Sx2 -12x-9

r 4x-7Applying L' Hospital rule, we get I = x~ lOx _ 12

4x3-7 5I = -1O-x-3---1-2= 18

26. The given differential equation may be written as02y + 4Dy + 3y = 3e2x

3e2x

.. EI. = (02 +40+3)

Substituting D = 2, we get

P.I. = 3e2xC1S) = e;x

27. Given, E;;; [~~~ ~~~ee8] and G;;; [~ ~ 8]o 0 1 001

Now Ex F = G

Px60Torque, T = -- = 23.87 N m27t x N

= Axial thrust, W = P x 7t (r12 - r;)

2 (r3 r3)T = - fl x P x 7t (r12 - r;) 12 - ~. = flwr

3 (rl - r2 )

Centre distance = D1 +D22

mTl +mT2 = S(19+37):"'140mm2 2

~Lcr=E-

L

200 x 103 ~L---- = 200x109x-(40)2 x 10-6 2

(J)max -(J)min(J)max + (J)min

2

«) 2+C fmax _

«)min - 2-C f

ForceStress =--

Area

328xlO N/m2

700 x 10-Q

(49-21)kN

Total potential energy =j(independent variable)Hence for a system in equilibirium,Total potential energy = ConstantThus, partial derivatives of its total potential energy with respect to each of independentvariable must be zero. TAB

TABsin60° + TACsin 30° = 600 (ii)

Solving equations (i) and (ii), we getTAB= 520N and TAC= 300N

(J) = {S = 20n VM

21t X N(J)= -- =37760

(J)2

:. Transmissibility ratio = 2 n 2 = 0.0028(J) -(J)n

41. According to Grashoff's Criteria.S+L::;P+Q

Strength of materialFactor of Safety = --------"--------

Actual load or strength on material

200(in MPa)15x 103

6 x 10-6 (inMPa)60 x --- x 10-6

cos45°

43. Given, u = x2t and v = - 2xyt

= 200(in MPa) = 3.458.91(in MPa)

a\jJ = v = -2xytOy

a\jJ 2- =-u=-x tOy

\jJ = -x2yt +f~y)Differentiating equation (iii) with respect to y, we get

8\jJ = _ x2t +fw)Oy

Equating the value of : from equations(ii) and (iv), we get

-x2t = - x2t +f' (y)Since,f~y) = 0, thus !(y) = C (where 'e' is constant of integration)

\jJ = -x2yt + CC is a numerical constant so it can be taken as zero

\jJ = -x2ytFor equation of stream lines, \jJ = constant

-x2yt = constantFor a particular instance, x2y = constant

44. By Hagen - Poiseuille law, for steady laminar flow in circular pipes

aut= -~- ar

-ap rax . 2

au aPr~ ar = ax'2

(-8r) P r~uo 02 = 1'2 ....u{ .."""o(1- ~:)]

-16~LuoP = 2o

45. By energy conservatioh, velocity at point Q == ~2g(h2 _ hI)

As there is a continuous and uniform flow, so velocity of liquid at point Q and P is same.p

Vp == ~2g(h2 -hI)

Heat generated by bulb == 100 x 24 x 60 x 60 Joule.

Heat dissipated == p x V x Cv x (T - 20)100 x 24 x 60 x 60 == 1.24 x 9 x 2.5 x 3 x C

v( T _ 20)

T == 4500

When added to Differential Function Phenomenonto the system

Heat Positive Inexact Path BoundaryWork Negative Inexact path Transient

57. a == (au + ~L au }t.L/L58. Let 'h' be height of mould and same will be liquid head.

cVTop gating,tA == ~2gh '

where, c = constantV = volume of mould

cxVBottom gating, tB == ~2g x 2h

Shear strain = cot~ + tan (~- a) == 1.65tA

tB == .fi

, Jti -tf1-1= --R

tf = 2.5 mm

Power generated = VI

25 x 300 x 0.85 _Heat input = Welding speed - 797 J/mm

63. Large negative rake angle, low shear angle and high cutting speed.

M1 M2

IN OUT IN OUT

R 0 8 8 21

T 8 19 21 35

S 19 46 35 55

Q 46 78 78 97

U 78 94 97 104

P 94 109 109 115Case I : Let EOQ is less than 500

~xCo. . EOQ = v- C

c= 447.21

Case II: Let EOQ is greater than 500

~2QXCo.. EOQ = C

c= 471.40 which is against the assumption.

. . EOQ = 447.21

. . . _ Total time used x 100Lme effIcIency - Number of work stations x Cycle time

48= --x 100 =80%

6xlOt1 0.5

Here, r = cutting ratio, n = t; = 0.7 = 0.714

rcosatan~= ---

l-rsinawhere, ~ = shear angleand ex = rake angle

~ = 40.2°Shear strain = cot ~ + tan (~- ex) = 1.65

From Merchant's theory,7t

2~ + A- ex = 2'A = friction angleA = 24.6

.. Il = tan A = 0.457Total heat generated, Q1 = Fe x V = 24000 J/ minHeat dissipated due to friction, Q2 = F x r xvFrictional force, F = Fe sin a + Ff cos a

= 503.76 NQ2 = 7193.69 J/min.

QzQz = 0.299 ; 30%

0)1 - 0)5=30)2 -0)5

0)3 -0)5=20)4 -0)5

0)1 -O)s= 60)4 -O)s

(J)1 = 60 rpm (Clockwise)(J)4 = 120 rpm (Counter clockwise)

60 - (J)s

-120- (J)s = 6(J)s = -156 i.e. counter clockwise

L wx2Bending Moment (B.M.) = w"2 x - -2-

where, w = weight/ unit lengthand L = length of rod

O)L2(B.M)max = -8-

L........ (x = "2)

Here, L = 6m, (J) = 1.5 kN / m. . (B.M)max= 8.75 kNm == 9 kNm

(Maximum load x distance) or Bending momentBending stress = Section modulus

= 9 x 103(Nm) = 211.26 MPa

~x(.075)332Hence closest option is 162.98 MPa

For critical damping, Cc = 2m x ~ = 223.6 Ns/m

Logarithmic decrement, 8 = ~ 21£X C = 0.42C2 _C2

c

Heat lost = n Cv drMass entering from side q-p = Mass leaving from side q- r + Mass leaving the side r-sBy momentum equation, we can find drag force.

Critical Path a - c - e - f30 + 60 + 45 + 20 = 155 days

85. Standard deviation, cr = .J25 +81 +36 + 9 days = .J151